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In this article, we consider the flat bundle   U   and the kernel   K   of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion    U  ⊆  K    can be in the geometric case. More precisely, for any smooth projective curve   C   of genus    g  ⩾  2    and any    r  =  0  ,  …  ,  g  −  1   , we construct non‐isotrivial deformations of   C   over a quasi‐projective base such that    rk  K  =  r    and    rk  U  ⩽    g  +  1   2    .

Families of curves with Higgs field of arbitrarily large kernel